Heat oil in a large nonstick skillet over medium-high heat until the oil should shimmer but not smoke. Add 3 1/2 cups of the cubed squash in an even layer and cook without stirring until golden brown, about 5 minutes. Season with salt and pepper and continue to cook, stirring occasionally, until the squash is tender and deeply browned, then transfer to a bowl.

Add the remaining cubed squash and the reserved squash fibers and seeds to the empty skillet and cook over medium heat, stirring often, until lightly browned, about 4 minutes. Transfer the mixture to a large saucepan and add the broth and water. Cover the pot, bring to a boil over high heat, then reduce the heat to medium-low and simmer gently for 10 minutes. Strain the broth through a fine-mesh strainer, pressing hard on the solids to extract as much flavorful liquid as possible, then return the strained broth to the saucepan and keep warm over low heat.

Melt 3 tablespoons of butter in the skillet over medium heat. Add the onions, salt, and pepper and cook until the onions are softened and translucent, 5 to 7 minutes. Stir in the garlic and cook until fragrant. Add the rice and cook, stirring frequently, until the edges of the grains begin to turn translucent, about 3 minutes.

Stir in the wine and cook, stirring constantly, until it is fully absorbed by the rice. Stir in 5 cups of the hot broth mixture and half of the browned squash cubes. Reduce the heat to medium-low, cover, and simmer until almost all the liquid has been absorbed and the rice is just al dente, 18 to 19 minutes, stirring only twice during this stage.

Add 3/4 cup of the hot broth mixture and stir gently and constantly until the risotto becomes creamy, about 3 minutes. Stir in the Parmesan, remove the pot from the heat, and cover for 5 minutes to let the rice finish. Finally, stir in the sage, nutmeg, and the remaining tablespoon of butter. Fold in the remaining browned squash cubes and adjust the consistency with the remaining broth if the rice seems too stiff.